Magnetic recording of transitions written onto a computer disc drive may produce an isolated pulse as shown below. This pulse comes from a disc drive's read/write channel. Each transition will cause such a signal to occur.

Figure 2.5.1 An isolated Readback Pulse from a disc drive

The signal's shape is very important to the electrical engineering development groups of disc drives. A readback pulse should be symmetric and have a relatively fast rise time (i.e. steep slope) for improved peak detection capability. A math model for the pulse can help gain insight into what electronic sub-system/components are causing the pulse to be asymmetric or have a slow rise time.

The longitudinal magnetic force was assumed the main contributing factor in determining a readback pulse shape, before the early 1980's. This force component was modeled by a series of three Lorentz functions. These functions have varying independent parameters that are dependent upon the drive's Thin-Film-Head (TFH) composition, size and shape. The value for these parameters was helpful in understanding a design and manufacturing flaw.

A Lorentz function has represented/modeled an isolated readback pulse for some time. The basic Lorentz function is defined as y = . The isolated pulse model is a composite of three Lorentz functions as stated here:

where
v_{i} = Amplitude of a Lorentz pulse; pw_50_{i} = Lorentz pulse width, measured at 50% height of v_{i}; and,
t_{i} = Origin of the ith Lorentz.

In the early 1980s, this model was found to be inadequate when Thin Film Heads were starting to be used in disc drives. An examination of the math model versus actual data plots showed that the 1970s model was no longer sufficient. The longitudinal force coupled with the increased vertical force were used to provide an excellent model for TFH readback pulses in the mid 1980s as shown in the following math model, signal_{2}:

where
v_{i} =
Amplitude of a longitudinal magnetic force;
vc_{i} = Amplitude of vertical force component;
pw_50_{i} = Lorentz pulse width, measured at 50% height of v_{i}; and,
t_{i} = Origin of the ith Lorentz
function.

Parameter Estimation Problem

Find the three Lorentz parameter set (v_{i},
vc_{i}, pw_50_{i} and t_{i})
values necessary to fit the Signal_{2}(t) model
to a digitized isolated readback pulse. Assume a total of 512 equal
spaced points (i.e. Δt constant between points).

Note: Our CurvFit
application will solve this curve fit problem with a Lorentz series
or a 'Modified' Lorentz series.

Increased Productivity Example #4
Source Code:

Problem Curvefit common v(3), vc(3), pw_50(3), time0(3), npoints, data(512), signal(512), error(512) data v/ -.1, 1., .05/, vc/ .1, .1, .1/, pw_50/ 50, 40, 50/ delta_t = .9765625 : npoints = 512 open( 11, file="pulse.dat", status=old) do j = 1, npoints read(11,*) data(j) signal(j) = (j - npoints/2) * delta_t end do FIND v, vc, pw_50, time0; IN Pulse; TO MATCH error ! 1. plot signal & data vs. time here ! 2. plot error vs. time here end Model Pulse common v(3), vc(3), pw_50(3), time0(3), npoints, data(512), signal(512), error(512) do j = 1, npoints sum = 0 do i = 1, 3 sum = sum + Lorentz( time(j), v(i), vc(i), pw_50(i), time0(i)) end do signal(j) = sum : error(j) = data(j) - signal(j) end do return end Function Lorentz( t, v_i, vc_i, pw_50_i, t_i) x = (t - t_i) / pw_50_i : num = v_i + vc_i * x den = 1 + x**2 : Lorentz = num / den return end

Increased Productivity Example #4
Other Information:

Relating Model and Design Parameters

Assuming the digitized data fits a math model with quadratic convergence, how do the model parameters (M) relate to the design dimensions? For example, this present TFH example has model parameters v_{i}, pw_50_{i}, and t_{i} (for i = 1 to 3) while the design parameters as shown in the following diagram are A, B, C, D, E, & F.

Figure 2.5.2 A TFH at flying height F above a disc drive's media

The governing equations may not be known for sure but someone with an understanding of the magnetic effects on a TFH could at least determine whether the parameters are proportional or inversely proportional. This would help as one starts building an understanding of what a math model might be in order to find the optimum design parameters to produce a symmetric and "narrow" (readback) pulse with no (or minimal) undershoots as represented in the curve shown below

Optimum pulse shape?

Figure 2.5.3 An "ideal" Readback Pulse from a disc drive

Through acquisition of many digitized pulses with varying pulse model parameters will eventually provide the necessary design parameters for an optimum pulse. This would require many man-hours of time. Another option would be to do Pulse Slimming
via a Matched Filter as Memorex Corp. did in the 1980s.

This Magnetic Recording problem is another increased productivity example do to using Calculus (level) programming.

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